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This paper studied the consensus problem of the leader-following multiagent system. It is assumed that the state information of the leader is only available to a subset of followers, while the communication among agents occurs at sampling instant. To achieve leader-following consensus, a class of distributed impulsive control based on sampling information is proposed. By using the stability theory of impulsive systems, algebraic graph theory, and stochastic matrices theory, a necessary and sufficient condition for fixed topology and sufficient condition for switching topology are obtained to guarantee the leader-following consensus of the multiagent system. It is found that leader-following consensus is critically dependent on the sampling period, control gains, and interaction graph. Finally, two numerical examples are given to illustrate the effectiveness of the proposed approach and the correctness of theoretical analysis.

During the past several decades, the consensus problem of the multiagent system has drawn a great deal of attentions because of its broad applications in many domains, including distributed coordination [

Inspired by some biological systems and engineering applications, the leader-following consensus problem has received a lot of interest. The leader is a special agent whose motion is independent of all other agents and thus is followed by all other agents. It has been widely used in many applications [

In recent years, owing to the development of digital sensors and the constraints of transmission bandwidth of networks, many control systems can be modeled by continuous-time systems together with discrete sampling. Therefore, it is significant to design the distributed control for continuous-time multiagent systems based on sampled information. There are a few reports [

On the other hand, impulsive dynamical systems exhibit continuous evolutions typically described by ordinary differential equations and instantaneous state jumps or impulses. It is also well known that the impulsive control is more efficient than one of continuous control in many situations. The examples include ecosystems management [

This paper aims to investigate the consensus problem of leader-following multiagent systems by using impulsive control which only regulates the velocity of agents. Our main contributions are summarised as follows. First, a necessary and sufficient condition under fixed topology is derived, and it is found that the leader-following consensus in multiagent systems with sampling information can be reached if and only if the sampled period is bounded by critical values which depend on control gains and the interaction graph. Second, a sufficient condition under switching topology is obtained, and it is shown that the impulsive interval is restricted by an upper bound which depends on control gains, the diagonal element of the Laplacian matrix, and the connections between agents and leader. The two key difference between this paper and our earlier work [

The remainingpart of the paper is organized as follows. In Section

Let

Let

Given a matrix

Consider that a multiagent system consists of

This paper focuses on the problem of designing

The leader-following consensus of the multiagent system (

In this section, the leader-following consensus problem under fixed topology is considered. The interaction between agents in this part is described by a fixed digraph

In order to achieve the leader-following consensus of the multiagent system (

Equivalently, the multiagent system (

From (

The multiagent system (

Let

Equivalently, (

The following lemmas and definition are needed for the subsequent development.

Polynomial

For complex polynomial

The complex polynomial

Next theorem will show what kind of interaction topology can reach leader-following consensus and how to determine the control gains

The multiagent system (

Let the

Let

Then, we only need to prove that polynomials

Let

It can be proved by Lemma

It can be observed from the inequality (

Let

How to choose a suitable control gain

In this section, the leader-following consensus under switching topology is considered. The interaction between agents at sampling time

In order to achieve leader-following consensus under switching topology, the impulsive control input is designed as

Note that the communication among agents only occurs at sampling times. This implies that interation graph does not contain any edges

Similar to the discussion in Section

It is easy to know that the network (

From (

Let

Before moving on, the following lemmas are needed.

Let

Let

Suppose that

Let

It is easy to check the non-negative matrix

Let

When

If there exists a positive integer

Let

If

If

then we have

If

then we have

Note that

The union of

By Lemma

According to Lemma

In this remark, we also show how to choose a suitable control gain

In this section, two illustrative numerical examples will be given to demonstrate the correctness of theoretical analysis.

The communication topology is described as in Figure

Fixed topology.

Trajectory of the multiagent system (

Trajectory of the multiagent system (

In this subsection, the network topology switches from a set

Switching topology:

Let

Trajectory of the multiagent system (

In this paper, the leader-following consensus problem of the multiagent system is considered. The impulsive control, which only needs sampled information and regulates the velocity of each agent at sampling times, is proposed for the leader-following consensus. Several new criteria are established for the leader-following consensus of the system under both fixed and switching topology. Illustrated examples have been given to show the effectiveness of the proposed impulsive control.

This work was supported in part by the National Natural Science Foundation of China under Grants 61073026, 61170031, 61272069, and 61073025.